Abstract

In this Letter, we describe the behavior of partially coherent, partially polarized focused vector beams after passing a linear polarizer placed at the focal plane of a high numerical aperture microscope lens. In particular, we develop a mathematical framework for such beams that helps the understanding of the performance of polarizers when interact with non-paraxial beams. The features of the focused field after the polarizer are numerically evaluated for some illustrative examples.

© 2018 Optical Society of America

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References

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  11. H. L. Ong, Jpn. J. Appl. Phys. 30, L1028 (1991).
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  21. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
    [Crossref]
  22. R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009), Vol. 147.
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  25. R. Martínez-Herrero and P. M. Mejías, Opt. Lett. 32, 1504 (2007).
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    [Crossref]
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    [Crossref]

2018 (1)

2017 (2)

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Opt. Lasers Eng. 98, 176 (2017).
[Crossref]

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Sci. Rep. 7, 42122 (2017).
[Crossref]

2016 (1)

2015 (1)

2013 (1)

2009 (3)

2007 (3)

2004 (1)

2002 (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[Crossref]

1993 (1)

1991 (2)

H. L. Ong, Appl. Phys. Lett. 59, 155 (1991).
[Crossref]

H. L. Ong, Jpn. J. Appl. Phys. 30, L1028 (1991).
[Crossref]

1984 (1)

1982 (1)

1980 (1)

P. Yeh, Surf. Sci. 96, 41 (1980).
[Crossref]

1979 (1)

1972 (1)

1959 (1)

B. Richards and E. Wolf, Proc. R. Soc. London A 253, 358 (1959).
[Crossref]

Aiello, A.

Banzer, P.

Berreman, D. W.

Cai, Y.

X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, J. Opt. Soc. Am. A 32, 2058 (2015).
[Crossref]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, Progress in Optics (Elsevier, 2017), Vol. 62, pp. 157–223.

Carnicer, A.

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Sci. Rep. 7, 42122 (2017).
[Crossref]

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Opt. Lasers Eng. 98, 176 (2017).
[Crossref]

Chen, Y.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, Progress in Optics (Elsevier, 2017), Vol. 62, pp. 157–223.

Fainman, Y.

Friberg, A. T.

Gbur, G.

G. Gbur and T. Visser, Progress in Optics (Elsevier, 2010), Vol. 55, pp. 285–341.

Gu, C.

Hellmann, C.

Juvells, I.

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Sci. Rep. 7, 42122 (2017).
[Crossref]

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Opt. Lasers Eng. 98, 176 (2017).
[Crossref]

Kaivola, M.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[Crossref]

Kolb, T.

Korger, J.

Leuchs, G.

Lindfors, K.

Liu, L.

X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, J. Opt. Soc. Am. A 32, 2058 (2015).
[Crossref]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, Progress in Optics (Elsevier, 2017), Vol. 62, pp. 157–223.

Liu, X.

X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, J. Opt. Soc. Am. A 32, 2058 (2015).
[Crossref]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, Progress in Optics (Elsevier, 2017), Vol. 62, pp. 157–223.

Maluenda, D.

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Opt. Lasers Eng. 98, 176 (2017).
[Crossref]

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Sci. Rep. 7, 42122 (2017).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Marquardt, C.

Martínez-Herrero, R.

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Sci. Rep. 7, 42122 (2017).
[Crossref]

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Opt. Lasers Eng. 98, 176 (2017).
[Crossref]

R. Martínez-Herrero and P. M. Mejías, Opt. Lett. 34, 2303 (2009).
[Crossref]

R. Martínez-Herrero and P. M. Mejías, Opt. Lett. 32, 1471 (2007).
[Crossref]

R. Martínez-Herrero and P. M. Mejías, Opt. Lett. 32, 1504 (2007).
[Crossref]

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009), Vol. 147.

Mejías, P. M.

Ong, H. L.

H. L. Ong, Appl. Phys. Lett. 59, 155 (1991).
[Crossref]

H. L. Ong, Jpn. J. Appl. Phys. 30, L1028 (1991).
[Crossref]

Partanen, H.

Piquero, G.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009), Vol. 147.

Richards, B.

B. Richards and E. Wolf, Proc. R. Soc. London A 253, 358 (1959).
[Crossref]

Santarsiero, M.

Setälä, T.

Shamir, J.

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[Crossref]

Tervo, J.

Visser, T.

G. Gbur and T. Visser, Progress in Optics (Elsevier, 2010), Vol. 55, pp. 285–341.

Wang, F.

Wittmann, C.

Wolf, E.

B. Richards and E. Wolf, Proc. R. Soc. London A 253, 358 (1959).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Wyrowski, F.

Yeh, P.

Yu, J.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, Progress in Optics (Elsevier, 2017), Vol. 62, pp. 157–223.

Zhang, S.

Zhao, C.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

H. L. Ong, Appl. Phys. Lett. 59, 155 (1991).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Jpn. J. Appl. Phys. (1)

H. L. Ong, Jpn. J. Appl. Phys. 30, L1028 (1991).
[Crossref]

Opt. Express (2)

Opt. Lasers Eng. (1)

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Opt. Lasers Eng. 98, 176 (2017).
[Crossref]

Opt. Lett. (5)

Phys. Rev. E (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[Crossref]

Proc. R. Soc. London A (1)

B. Richards and E. Wolf, Proc. R. Soc. London A 253, 358 (1959).
[Crossref]

Sci. Rep. (1)

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Sci. Rep. 7, 42122 (2017).
[Crossref]

Surf. Sci. (1)

P. Yeh, Surf. Sci. 96, 41 (1980).
[Crossref]

Other (5)

P. Yeh, Optical Waves in Layered Media (Wiley, 1988), Vol. 95.

G. Gbur and T. Visser, Progress in Optics (Elsevier, 2010), Vol. 55, pp. 285–341.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, Progress in Optics (Elsevier, 2017), Vol. 62, pp. 157–223.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009), Vol. 147.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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Figures (4)

Fig. 1.
Fig. 1. Coordinate system.
Fig. 2.
Fig. 2. Non-polarized coherent beam a=b=1, NA=0.9, f0=1, and β=0. The minimum value of P3D is 0.52. These distributions are calculated at the plane xy (see Fig. 1). Axis values in λ units.
Fig. 3.
Fig. 3. Almost incoherent beam (Lc/f=0.3), NA=0.9, f0=1, β=0: (a) radially polarized (a=0,b=1), min(P3D)=0.51; (b) azimuthally polarized (a=1,b=0), min(P3D)=0.64. These distributions are calculated at the plane xy (see Fig. 1). Axis values in λ units.
Fig. 4.
Fig. 4. Partially coherent beam (Lc/f=3), NA=0.9, f0=1, β=0: (a) radially polarized (a=0,b=1), min(P3D)=0.5; (b) azimuthally polarized (a=1,b=0), min(P3D)=0.5. These distributions are calculated at the plane xy (see Fig. 1). Axis values in λ units.

Equations (22)

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E(r)=A0θM02πcosθE0(θ,ϕ)exp(ikr·s)sinθdθdϕ,
E(r)=A0θM02πcosθ(E0(θ,ϕ)·q(θ,ϕ,β))p(θ,ϕ,β)×exp(ikr·s)sinθdθdϕ,
q(θ,ϕ,β)=ts(θ)cosθ0cos(ϕβ)1sin2θ0cos2(ϕβ)e1tp(θ)sin(ϕβ)1sin2θ0cos2(ϕβ)e2,
p(θ,ϕ,β)=ts(θ)cosθ0cos(ϕβ)1sin2θ0cos2(ϕβ)e1+tp(θ)sin(ϕβ)1sin2θ0cos2(ϕβ)e2,
ts(θ)=2cosθcosθ+nocosθ0tp(θ)=2cosθcosθ0+nocosθ,
ts(θ)=2nocosθ0cosθ+nocosθ0tp(θ)=2nocosθ0cosθ0+nocosθ.
W^p(r1,r2)=|A|20θM0θM02π02πW^o(θ1,ϕ1,θ2,ϕ2)×exp(ik(r1s1r2s2))sinθ1sinθ2dθ1dθ2dϕ1dϕ2,
W^o(θ1,ϕ1,θ2,ϕ2)=cosθ1cosθ2Γ(θ1,ϕ1,θ2,ϕ2)×p(θ1,ϕ1,β)p(θ2,ϕ2,β),
Γ(θ1,ϕ1,θ2,ϕ2)=q(θ1,ϕ1)W^i(θ1,ϕ1,θ2,ϕ2)q(θ2,ϕ2).
W^i(θ1,ϕ1,θ2,ϕ2)=nλn2Fn(θ1,ϕ1)Fn(θ2,ϕ2).
W^P(r1,r2)=nλn2Hn(r1)Hn(r2),with
Hn(r)=0θM02πcosθ(Fn(θ,ϕ)·q(θ,ϕ))p(θ,ϕ)exp(ikrs)sinθdθdϕ.
P3D2=32(Tr[W^p(r,r)W^p(r,r)][Tr[W^p(r,r)]]213),
P3D2=13nmλn2λm2|Hn(r)|2|Hm(r)|2sin2αnm(r)2(nλn2|Hn(r)|2)2,
sin2αnm(r)=1|Hn(r)Hm(r)|2|Hn(r)|2|Hm(r)|2.
W^i(θ1,ϕ1,θ2,ϕ2)=Γi(θ1,ϕ1,θ2,ϕ2)M^,
|μW|max=|Γi(θ1,ϕ1,θ2,ϕ2)|Γi(θ1,ϕ1,θ1,ϕ1)Γi(θ2,ϕ2,θ2,ϕ2).
P3D2=13ab|H1(r)|2|H2(r)|2sin2α12(r)(a|H1(r)|2+b|H2(r)|2)2,
Hj(r)=A0θM02πcosθ(qU)j*g(θ,ϕ)p(θ,ϕ)×exp(ikrs)sinθdθdϕ,
P3Dj2=13nmλn2λm2|Hnj(r)|2|Hmj(r)|2sin2αnmj(r)2(nλn2|Hnj(r)|2)2,
Hnj(r)=A0θM02πcosθ(qU)j*gn(θ,ϕ)p(θ,ϕ)×exp(ikrs)sinθdθdϕ,
Γi(θ1,ϕ1,θ2,ϕ2)sinθ1sinθ2fo2  sin2θMexp(sin2θ1+sin2θ2fo2sin2θM)×exp(sin2θ1+sin2θ22sinθ1sinθ2cos(ϕ1ϕ2)Lc2/f2)×exp(i(ϕ2ϕ1)).